Ellipsometric characterization of inhomogeneous thin films with complicated thickness non-uniformity: application to inhomogeneous polymer-like thin films

Ivan Ohlídal; Jiří Vohánka; Vilma Buršíková; Václav Šulc; Štěpán Šustek; Miloslav Ohlídal

doi:10.1364/OE.412043

1. Introduction

Thin films play an important role in many branches of fundamental and applied research. They are also widely utilized in optical investigations and applications. Both the homogeneous and inhomogeneous thin films are encountered in practice. The homogeneous thin films have the same optical constants within their volumes while the optical constants change within the volumes of the inhomogeneous thin films. These changes mostly correspond to continuous profiles of the optical constants along the axis perpendicular to their boundaries. An enormous number of papers concerning the characterization of the homogeneous thin films has been published (see e.g. [1–12]). A less attention has been devoted to the optical characterization of the inhomogeneous thin films. This is caused by more complicated theoretical approaches needed for the complete optical characterization of these thin films. However, in the several last years the number of papers dealing with the optical characterization of the inhomogeneous thin films increased (see e.g. [13–19]). This is caused by an increased utilization of thin films formed by complex materials, which often exhibit the optical inhomogeneity, in various branches of applied research, optics industry, semiconductor technology, solar energetics, microelectronics etc. The non-stoichiometric silicon nitride films that are rich in silicon (see e.g. [15]) and some films prepared by plasma-chemical deposition (see e.g. [20]) are typical examples. Moreover, the inhomogeneous thin films find increased utilization in miscellaneous devices produced in optics industry. The typical representatives of such the inhomogeneous thin films are rugate filters employed as the optical devices with high reflectance in certain spectral ranges (see e.g. [21]).

Many homogeneous and inhomogeneous thin films exhibit various defects, which influence their optical properties. If the defects are not taken into account in the optical characterization of these films, the obtained results could be misleading. There are many papers devoted to the optical characterization of the homogeneous thin films containing various defects. The homogeneous thin films with randomly rough boundaries were optically characterized, for example, in papers [22–32]. The influence of thickness non-uniformity on the optical characterization of the homogeneous thin films was studied in papers [33–40]. In paper [40] the thickness non-uniformity of the homogeneous thin films is expressed by means of the model utilizing polynomials to describe the distribution of local thicknesses along the surface of the sample. This approach is general in principle and, moreover, it allows to characterize the films with the complicated thickness non-uniformity. In papers [33,35,36] the thickness non-uniformity is described by a simple model corresponding to the wedge-shaped non-uniformity, which leads to the distribution of local thicknesses given by the Wigner semicircle distribution. In papers [34], the thickness non-uniformity is described by the simplest model of the local thicknesses uniformly distributed in a certain interval, which represents a naïve model of the thickness non-uniformity with a limited utilization in practice. Imaging spectroscopic reflectometry (ISR) has been used to characterize the non-uniform homogeneous thin films in papers [37–39]. Within this technique it is not necessary to assume a model of thickness non-uniformity, because the non-uniformity is characterized directly if the optical constants of the films are known or determined independently. The optical characterization of homogeneous thin films containing transition layers was performed, for instance, in [41,42]. The complete optical characterization of the homogeneous epitaxial ZnSe thin films exhibiting the combined defect consisting of random roughness of the upper boundaries, thickness non-uniformity corresponding to the wedge-shaped model and overlayers occurring on their upper boundaries was performed using the combined method of spectroscopic ellipsometry and spectroscopic reflectometry in paper [43].

So far the influence of defects on the optical characterization of the inhomogeneous thin films has not been studied in a sufficient way. To our knowledge there are three papers dealing with the complete optical characterization of the inhomogeneous thin films having defects. In [20] the complete optical characterization of inhomogeneous non-stoichiometric silicon nitride (SiN$_x$) thin films and inhomogeneous polymer-like (SiO$_x$C$_y$H$_z$) thin films exhibiting homogeneous transition layers at the lower boundaries was performed using the combined method of spectroscopic ellipsometry and spectroscopic reflectometry. The optical properties of the transition layers were determined simultaneously with the optical properties of the inhomogeneous SiN$_x$ and SiO$_x$C$_y$H$_z$ thin films. In [44] the inhomogeneous thin SiN$_x$ films exhibiting random roughness of the upper boundaries and unwanted uniaxial optical anisotropy inside their volumes were completely characterized from the optical point of view.

The complete optical characterization of the inhomogeneous SiO$_x$C$_y$H$_z$ thin films exhibiting thickness non-uniformity and transition layers at their lower boundaries was carried out in paper [45]. In this paper, the wedge-shaped model was used to describe the thickness non-uniformity. In papers [20,45], the influence of inhomogeneity on the reflection coefficients of the SiO$_x$C$_y$H$_z$ thin films was expressed on the basis of the multiple-beam interference model derived in [46].

Some inhomogeneous SiO$_x$C$_y$H$_z$ thin films prepared using plasma enhanced chemical vapor deposition (PECVD) under certain technological conditions exhibited a complicated thickness non-uniformity. The simple wedge-shaped model could not be used to characterize these films. A more general method expressing the thickness non-uniformity by means of polynomials in coordinates along the surfaces of the films had to be used. In this paper, the method is illustrated by the complete optical characterization of a selected sample of the inhomogeneous SiO$_x$C$_y$H$_z$ thin film exhibiting more complicated thickness non-uniformity than the films optically characterized in our recent paper [45].

2. Sample preparation and experimental arrangement

2.1 Sample preparation

The samples of the inhomogeneous thin films of SiO$_x$C$_y$H$_z$ exhibiting thickness non-uniformity were prepared using PECVD onto silicon single crystal substrates. The deposition was performed using a parallel plate reactor with capacitively coupled glow discharge at working frequency of 13.56 MHz. The reaction chamber was made of a glass cylinder closed by two stainless steel flanges and the parallel electrodes were made of graphite. The RF power was applied to the lower electrode together with a negative DC self bias voltage induced in order to control the acceleration of ions bombarding the growing film. Prior the deposition of these films, the silicon substrates were pretreated for 5 minutes in argon (Ar) discharge at applied power 50 W, bias voltage −320 V and flow rate of argon 5 sccm. The film was deposited using the mixture of methane (CH$_4$) and hexamethyldisiloxane (C$_6$H$_{18}$Si$_2$O - HMDSO) supplied into the reactor chamber by the glass torus with many outlets on its perimeter. The flow of HMDSO was maintained at 2 sccm. In order to vary the composition of SiO$_x$C$_y$H$_z$ films, the methane flow rate was gradually reduced from 5.5 sccm to 0 sccm for 5 minutes. The supplied power was 50 W and the negative self bias voltage on the bottom electrode was −80 V at the beginning of the deposition and it was −100 V at its end. The pressure at the beginning of the deposition was 34.5 Pa and at the end it was 15.8 Pa. It should be noted that the samples were placed on the metal sample holder during deposition. In order to create films with pronounced thickness non-uniformity, we deliberately positioned the samples near the edges of this holder where the growth of the films is influenced by the distorted electric field. The characterized thin films of SiO$_x$C$_y$H$_z$ were deposited onto the unheated silicon substrates (wafers) with temperature about 30$^{\circ }$C.

2.2 Experimental arrangement

A Horiba Jobin Yvon UVISEL phase modulated ellipsometer was used to measure the associated ellipsometric parameters. The spectral dependencies of these ellipsometric parameters were measured for five incidence angles in the interval 55–75$^{\circ }$ within the spectral range 0.6–6.5 eV (190–2066 nm). Four orientations of the samples owing to the plane of incidence of light were utilized in these ellipsometric measurements (see below). ISR was used to measure local reflectance maps of the prepared inhomogeneous non-uniform SiO$_x$C$_y$H$_z$ thin films. The reflectance maps enable us to determine the maps of the local thicknesses of these films. These measurements were carried out by the original imaging spectroscopic photometer described in detail in [47]. The technique of ISR was employed within the spectral range 1.8–4.5 eV (275–700 nm) at normal incidence of light. One pixel corresponded to area of 80$\times$80 $\mu$m on the sample surfaces.

3. Theoretical background

3.1 Structural model

The structural model of the inhomogeneous non-uniform thin films characterized in this paper (see Fig. 1) is specified by the following assumptions:

1. The thin film consists of isotropic absorbing material from the optical point of view.
2. The profiles of the optical constants are described by continuous complex functions of one variable corresponding to the z-axis.
3. The thin film exhibits a thickness non-uniformity, i.e. local thickness of this film changes along substrate.
4. At the lower boundary of the film a uniform homogeneous transition layer consisting of isotropic absorbing material occurs.
5. The substrate is optically absorbing homogeneous and isotropic.
6. Boundaries of the thin film and transition layer are smooth.
7. The ambient is formed by a non-absorbing homogeneous isotropic material.
8. No defects occur in volumes of the thin film, transition layer and substrate.

Fig. 1. A schematic diagram of the structural model (left). A schematic diagram of four orientations of the sample owing to the plane of incidence (right), the dashed line represents the plane of incidence.

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It is assumed that the optical constants of the thin film specified above are dependent on coordinate $z$ through dispersion parameters $p_\alpha (z)$ expressed by the following function:

(1)$$p_\alpha(z) = p^\textrm{L}_\alpha + ( p^\textrm{U}_\alpha - p^\textrm{L}_\alpha ) \frac{z}{h(x,y)}$$

where symbols $p^\textrm {U}_\alpha$ and $p^\textrm {L}_\alpha$ represent the dispersion parameters at upper and lower boundaries, respectively. Symbols $x$ and $y$ represent coordinates along the surfaces of the films with the origin chosen at the center of the light spot of the ellipsometer. The local thickness values $h(x,y)$ are expressed as follows:

(2)$$h(x,y) = h_0 + h_\textrm{x} \frac{x}{R} + h_\textrm{y} \frac{y}{R} + h_\textrm{xx} \frac{x^{2}}{R^{2}} + h_\textrm{xy} \frac{xy}{R^{2}} + h_\textrm{yy} \frac{y^{2}}{R^{2}} + \cdots$$

where $h_0$, $h_\textrm {x}$, $h_\textrm {y}$, $h_\textrm {xx}$, $h_\textrm {xy}$, $h_\textrm {yy}$, etc. are parameters of the model of non-uniformity. The normalization owing to beam radius $R$ of incident light is used because the value of this quantity cannot be determined using the ellipsometric measurements.

3.2 Dispersion model

The optical constants of the inhomogeneous non-uniform thin film are described by the Campi–Coriasso model [48,49]. In this model the imaginary part of the dielectric function $\varepsilon _\textrm {i}(E)$ of the complex dielectric function is given as follows:

(3)$$\varepsilon_\textrm{i}(E) = \frac{2 N_\textrm{vc}}{\pi E} \frac{ B (E-E_\textrm{g})^{2} \Theta(E-E_\textrm{g}) }{ [(E_\textrm{c}-E_\textrm{g})^{2} - (E-E_\textrm{g})^{2}]^{2} + B^{2} (E-E_\textrm{g})^{2} }$$

where $E$ is photon energy, $N_\textrm {vc}$ denotes the strength of the interband transitions, $\Theta (\cdot )$ is the Heaviside function, $E_\textrm {g}$ represents the band gap energy. The parameters $B$ and $E_\textrm {c}$ describe the position and width of the absorption peak ($E_\textrm {c} > E_\textrm {g}$). The real part of the complex dielectric function $\varepsilon _\textrm {r}(E)$ is calculated by means of the Kramers–Kronig relation. The dielectric response of the transition layer is expressed by the combination of two Campi–Coriasso terms and exponential tail.

3.3 Reflection coefficients of inhomogeneous non-uniform thin films

The reflection coefficients of the inhomogeneous thin films exhibiting thickness non-uniformity can be expressed using the modification of the formulae for the inhomogeneous thin films based on the multiple-beam interference model presented in our earlier paper [46]. This modification consists in replacing the constant thickness by the variable local thickness expressed by the function $h(x,y)$ in the formulae for the reflection coefficients of the inhomogeneous thin films. This means that the formulae for the reflection coefficients of the inhomogeneous thin films exhibiting the thickness non-uniformity are expressed as

(4)$$r_q = r_{0q} + \sum_{n=1}^{D} \Delta r_{nq} ,$$

where $q=\textrm {p},\textrm {s}$ and

(5)$$\begin{aligned} r_{0q} &= \frac{ r_{\textrm{U}q} + r_{\textrm{L}q} \textrm{e}^{\textrm{i}x_\textrm{d}} }{ 1 + r_{\textrm{U}q} r_{\textrm{L}q} \textrm{e}^{\textrm{i}x_\textrm{d}} } , & x_\textrm{d} &= \frac{4\pi}{\lambda} \int_0^{h(x,y)} \sqrt{ n^{2}(z) - n_0^{2} \sin^{2}\phi_0 } \, \textrm{d}z ,\\ \Delta r_{nq} &= \sum_{l=1}^{3^{n-1}} \left( I^{(l)}_{nq} + \overline{I}^{(l)}_{nq} \right), \end{aligned}$$

Symbols $\lambda$, $n_0$ and $\phi _0$ denote the wavelength, refractive index of the ambient and incidence angle on the upper boundaries, respectively. Symbol $n(z)$ represents the complex function describing the profile of the complex refractive index of these films. Furthermore, it holds:

(6)$$\begin{aligned} r_{\textrm{U}q} &= \frac{ Y_{0q} - Y_{\textrm{U}q} }{ Y_{0q} + Y_{\textrm{U}q} }, & r_{\textrm{L}q} &= \frac{ Y_{\textrm{L}q} - Y_{\textrm{S}q} }{ Y_{\textrm{L}q} + Y_{\textrm{S}q} },\\ Y_{0\textrm{s}} &= n_0 \cos\phi_0, & Y_{0\textrm{p}} &= \frac{n_0}{\cos\phi_0}, & Y_{\textrm{U}\textrm{s}} &= n_\textrm{U} \cos\phi_\textrm{U}, & Y_{\textrm{U}\textrm{p}} &= \frac{n_\textrm{U}}{\cos\phi_\textrm{U}},\\ Y_{\textrm{L}\textrm{s}} &= n_\textrm{L} \cos\phi_\textrm{L}, & Y_{\textrm{L}\textrm{p}} &= \frac{n_\textrm{L}}{\cos\phi_\textrm{L}}, & Y_{\textrm{S}\textrm{s}} &= n_\textrm{S} \cos\phi_\textrm{S}, & Y_{\textrm{S}\textrm{p}} &= \frac{n_\textrm{S}}{\cos\phi_\textrm{S}}, \end{aligned}$$

Symbols $Y_{0\textrm {p}}$, $Y_{\textrm {U}\textrm {p}}$, $Y_{\textrm {L}\textrm {p}}$ and $Y_{\textrm {S}\textrm {p}}$ represent the admittances of the ambient, at the upper boundaries of the films, at the lower boundaries of the films and the substrates for the p polarization, respectively. Symbols $Y_{0\textrm {s}}$, $Y_{\textrm {U}\textrm {s}}$, $Y_{\textrm {L}\textrm {s}}$ and $Y_{\textrm {S}\textrm {s}}$ denote the corresponding admittances for the s polarization. Symbols $n_\textrm {U}$, $n_\textrm {L}$ and $n_\textrm {S}$ denote the refractive indices at the upper boundaries of the films, refractive indices at the lower boundaries of the films and refractive indices of the substrates, respectively. Symbols $\phi _\textrm {U}$, $\phi _\textrm {L}$ and $\phi _\textrm {S}$ are the refraction angles at the upper boundaries, at the lower boundaries and substrates, respectively. It is evident that the following equations are true:

(7)$$\begin{aligned} \cos\phi_\textrm{U} &= \frac{1}{n_\textrm{U}} \sqrt{ n_\textrm{U}^{2} - n_0^{2} \sin^{2} \phi_0 } , & \cos\phi_\textrm{L} &= \frac{1}{n_\textrm{L}} \sqrt{ n_\textrm{L}^{2} - n_0^{2} \sin^{2} \phi_0 } ,\\ \cos\phi_\textrm{S} &= \frac{1}{n_\textrm{S}} \sqrt{ n_\textrm{S}^{2} - n_0^{2} \sin^{2} \phi_0 } , \end{aligned}$$

Note that $n_0$, $\phi _0$ and $h(x,y)$ are always real quantities while the other quantities occurring in the foregoing equations are complex quantities in general. Symbol $\Delta r_{nq}$ denotes the total correction of the $n$-th order. Symbols $I^{(l)}_{nq}$ and $\overline {I}^{(l)}_{nq}$ represent the partial corrections of the $n$-th order (for details see [46]). The maximum order $D$ is given by an accuracy needed for calculating optical quantities expressed by means of the reflection coefficients $r_q$. The formulae for $r_{0q}$ correspond to the Wentzel–Kramers–Brillouin–Jeffreys (WKBJ) approximation of the reflection coefficients valid for the inhomogeneous thin films with negligible gradients of the refractive index profiles. In the following formulae except (17) and (18) the lower index $q$ will be omitted. Then the total corrections of the first order are expressed as follows (see [45,46]):

(8)$$\Delta r_1 = I_1^{(1)} + \overline{I}_1^{(1)} ,$$

where

(9)$$\begin{aligned} I_1^{(1)} &= C_1^{(1)} j_1^{(1)}, & C_1^{(1)} &= \frac{ \tau \textrm{e}^{\textrm{i} x_\textrm{d}} }{ (1 - \rho)^{2} }, & j_1^{(1)} &= \int_0^{h(x,y)} f(z_1) \textrm{e}^{-\textrm{i}x(z_1)} \, \textrm{d}z_1 ,\\ \overline{I}_1^{(1)} &= \overline{C}_1^{(1)} \overline{j}_1^{(1)}, & \overline{C}_1^{(1)} &= - \frac{ \tau r_\textrm{L} Z }{ (1 - \rho)^{2} }, & \overline{j}_1^{(1)} &= \int_0^{h(x,y)} f(z_1) \textrm{e}^{\textrm{i}x(z_1)} \, \textrm{d}z_1 ,\\ Z &= r_\textrm{L} \textrm{e}^{\textrm{i}x_\textrm{d}} , & f(z) &= \frac{1}{2 Y(z)} \frac{\textrm{d}Y(z)}{\textrm{d}z} , & x(z_1) &= \frac{4\pi}{\lambda} \int_0^{z_1} \sqrt{ n^{2}(z) - n_0^{2} \sin^{2}\phi_0 } \, \textrm{d}z ,\\ \rho &= - r_\textrm{U} r_\textrm{L} \textrm{e}^{\textrm{i}x_\textrm{d}} , & \tau &= t_\textrm{U} t'_\textrm{U} , \end{aligned}$$

The transmission coefficients of the upper boundaries of the films $t_\textrm {U}$ and $t'_\textrm {U}$ are given as

(10)$$t_\textrm{U} = \frac{ 2 Y_0 }{ Y_0 + Y_\textrm{U} }, \quad t'_\textrm{U} = \frac{ 2 Y_\textrm{U} }{ Y_0 + Y_\textrm{U} },$$

for the s polarization

(11)$$t_\textrm{U} = \frac{\cos\phi_0}{\cos\phi_\textrm{U}} \frac{ 2 Y_0 }{ Y_0 + Y_\textrm{U} }, \quad t'_\textrm{U} = \frac{\cos\phi_\textrm{U}}{\cos\phi_0} \frac{ 2 Y_\textrm{U} }{ Y_0 + Y_\textrm{U} },$$

for the p polarization.

The total corrections of the second and higher orders of the inhomogeneous non-uniform thin films are given by the formulae presented in our paper [46]. However, one change must be performed in these formulae. The local thicknesses expressed by (2) must be inserted into the upper bounds of the integrals occurring in these formulae (see (8) for the partial corrections of the first order). The formulae presented in our paper [46] must be rearranged for calculations in this paper because in paper [46] the opposite orientation of axis $z$ was employed. If transition layers occur at the lower boundaries of the inhomogeneous non-uniform thin films the reflection coefficients of their lower boundaries $r_\textrm {L}$ have to be expressed as follows:

(12)$$r_\textrm{L} = \frac{ r_\textrm{LT} + r_\textrm{TS} \textrm{e}^{\textrm{i}x_\textrm{T}} }{ 1 + r_\textrm{LT} r_\textrm{TS} \textrm{e}^{\textrm{i}x_\textrm{T}} } ,$$

where

(13)$$\begin{aligned} r_\textrm{LT} &= \frac{ Y_\textrm{L} - Y_\textrm{T} }{ Y_\textrm{L} + Y_\textrm{T} } , & r_\textrm{TS} &= \frac{ Y_\textrm{T} - Y_\textrm{S} }{ Y_\textrm{T} + Y_\textrm{S} } , & x_\textrm{T} &= \frac{4\pi}{\lambda} d_\textrm{T} n_\textrm{T} \cos\phi_\textrm{T} . \end{aligned}$$

Symbols $Y_\textrm {T}$, $n_\textrm {T}$ and $d_\textrm {T}$ denote the admittances, refractive indices and thicknesses of the transition layers, respectively ($Y_\textrm {T}$ and $n_\textrm {T}$ are complex quantities in general). The values of $n_\textrm {U}$, $\phi _\textrm {U}$, $Y_\textrm {U}$, $r_\textrm {U}$ and dispersion parameters at the upper boundaries are independent on the values of the local thickness. Thus, these quantities do not change along the upper boundaries. On the contrary, the gradients of the refractive index profiles are dependent on the local thickness values. This means that the values of the refractive index are different in the same heights above the lower boundaries of such the films.

3.4 Ellipsometric parameters of the inhomogeneous non-uniform thin films

The thickness non-uniformity of thin films can be described using the Stokes–Mueller formalism. The Stokes–Mueller formalism is described, for example, in [50,51]. The associated ellipsometric parameters are elements of the normalized Mueller matrix (see e.g. [50,51]). For reflected light these parameters are given by the following Mueller matrix:

(14)$$M = R_0 \begin{pmatrix} 1 & -I_\textrm{n} & 0 & 0 \\ -I_\textrm{n} & 1 & 0 & 0 \\ 0 & 0 & I_\textrm{c} & I_\textrm{s} \\ 0 & 0 & -I_\textrm{s} & I_\textrm{c} \end{pmatrix} ,$$

where $R_0$ denotes the reflectance of the samples and $I_\textrm {n}$, $I_\textrm {s}$ and $I_\textrm {c}$ are the individual associated ellipsometeric parameters. The degree of polarization $P$ describing a depolarization of light reflected from the samples is defined by means of these ellipsometric parameters as follows:

(15)$$P = \sqrt{ I_\textrm{s}^{2} + I_\textrm{c}^{2} + I_\textrm{n}^{2} } .$$

The Mueller matrix of the thin films exhibiting area non-uniformity in thickness along substrates is expressed as follows (see e.g. [52,53]):

(16)$$\overline{M} = \frac{1}{S} \iint_S M(h(x,y)) \,\textrm{d}S ,$$

where the integration is performed over the light spot illuminated on the sample, $S$ is the area of this spot and $M(h(x,y))$ denotes the Mueller matrix corresponding to a point on the surface of the sample with the local thickness $h(x,y)$. Note that formula (16) is true if the illumination by incident beam is constant over the entire light spot.

The associated ellipsometric parameters $I_\textrm {c}$, $I_\textrm {n}$ and $I_\textrm {s}$ of the inhomogeneous non-uniform thin films having the transition layers are calculated using the local reflection coefficients $r_\textrm {p}$ and $r_\textrm {s}$ in the following way (see e.g. [36,45,51,53]):

(17)$$I_\textrm{s} = \textrm{i} \frac{ \left\langle r_\textrm{p} r_\textrm{s}^{*} \right\rangle - \left\langle r_\textrm{p}^{*} r_\textrm{s} \right\rangle }{ \left\langle |r_\textrm{s}|^{2} \right\rangle + \left\langle |r_\textrm{p}|^{2} \right\rangle } , \quad I_\textrm{c} = - \frac{ \left\langle r_\textrm{p} r_\textrm{s}^{*} \right\rangle + \left\langle r_\textrm{p}^{*} r_\textrm{s} \right\rangle }{ \left\langle |r_\textrm{s}|^{2} \right\rangle + \left\langle |r_\textrm{p}|^{2} \right\rangle } , \quad I_\textrm{n} = \frac{ \left\langle |r_\textrm{s}|^{2} \right\rangle - \left\langle |r_\textrm{p}|^{2} \right\rangle }{ \left\langle |r_\textrm{s}|^{2} \right\rangle + \left\langle |r_\textrm{p}|^{2} \right\rangle } ,$$

where the angle brackets denote the mean values of the corresponding quantities calculated as follows:

(18)$$\begin{aligned} \left\langle r_\textrm{p} r_\textrm{s}^{*} \right\rangle &= \frac{1}{S} \iint_S r_\textrm{p} r_\textrm{s}^{*} \,\textrm{d}S , & \left\langle r_\textrm{p}^{*} r_\textrm{s} \right\rangle &= \frac{1}{S} \iint_S r_\textrm{p}^{*} r_\textrm{s} \,\textrm{d}S ,\\ \left\langle |r_\textrm{s}|^{2} \right\rangle &= \frac{1}{S} \iint_S |r_\textrm{s}|^{2} \,\textrm{d}S , & \left\langle |r_\textrm{p}|^{2} \right\rangle &= \frac{1}{S} \iint_S |r_\textrm{p}|^{2} \,\textrm{d}S , \end{aligned}$$

The reflection coefficients $r_\textrm {p}$ and $r_\textrm {s}$ are expressed by Eq. (4).

3.5 Numerical calculations of the integrals

The integration concerning inhomogeneity was performed using the numerical method based on the Chebyshev interpolation. The integrands of the corresponding integrals were approximated by the polynomials and the integrals of these polynomials were then calculated analytically. The details of this method are described in our earlier paper [46]. The numerical integration corresponding to the thickness non-uniformity was performed using the method of the Gaussian quadrature. The applied procedure of this method is described in detail in our recent paper [40]. The combination of these two numerical methods was very efficient for the processing of the experimental data obtained for the SiO$_x$C$_y$H$_z$ thin films exhibiting the complicated structure.

4. Data processing

The ellipsometric measurements are sensitive mostly to the shape of thickness non-uniformity along the direction lying in the incidence plane where the light spot is the longest. Therefore, we measured the ellipsometric parameters for four orientations of the samples owing to the plane of incidence to determine the thickness non-uniformity in a more detailed way. These orientations are illustrated using a schematic diagram shown in Fig. 1. The studied samples were covered by a mask with a circular hole so that selected parts of the inhomogeneous non-uniform thin films characterized were easily identified. The experimental spectral dependencies of the ellipsometric parameters were processed simultaneously for all the incidence angles and all the rotations corresponding to the individual orientations (these rotations corresponded to azimuth angles $\beta =0^{\circ }$, $45^{\circ }$, $90^{\circ }$ and $135^{\circ }$, see Fig. 1). The least-squares method was employed for this processing.

Fig. 2. Spectral dependencies of the optical constants $n_\textrm {T}$ and $k_\textrm {T}$ of the transition layer. Symbols $n$ and $k$ represent refractive index and extinction coefficient of silicon single crystal [54] (left). Spectral dependencies of the associated ellipsometric parameters at angle of incidence of 70$^{\circ }$ (right): points denote the experimental values, curves denote the theoretical data.

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The technique of ISR was utilized for confirming the results concerning the thickness non-uniformity determined by ellipsometry. Thus, this technique was used to determine the geometry of the surface corresponding to the upper boundary of the inhomogeneous non-uniform thin films characterized. The spectral dependencies of the local reflectance measured by individual pixels at normal incidence were processed using the formulae for uniform thin films because it was possible to assume that the characterized films were uniform in thickness within the areas on these films corresponding to the individual pixels. The detailed description of ISR and its applications is presented, for example, in [47].

5. Results and discussion

The described ellipsometric method for the complete optical characterization of inhomogeneous thin films with the complicated or considerable thickness non-uniformity is illustrated using one of the characterized thin films of SiO$_x$C$_y$H$_z$.

In addition to the sample with SiO$_x$C$_y$H$_z$ thin film, the silicon single crystal substrate pretreated in Ar discharge was optically characterized. In this way one could reliably obtain the values of thickness and optical constants of the transition layer originating in consequence of this pretreating of the silicon substrate. A native oxide layer was taken into account on top of the transition layer. The spectral dependencies of the optical constants of the native oxide layer was fixed as usual in values corresponding to amorphous SiO$_2$. Using the processing of the experimental data the thickness values of the native oxide layer $d_0$ and transition layer $d_\textrm {T}$ were determined in the following values: $d_0=2.9 \pm 0.2$ nm and $d_\textrm {T}=12.4 \pm 0.2$ nm. Moreover, the spectral dependencies of the refractive index $n_\textrm {T}$ and extinction coefficient $k_\textrm {T}$ were determined (see Fig. 2). It is apparent that the spectral dependencies of the optical constants of the transition layer are strongly different from those corresponding to silicon single crystal. It is known that the values of the optical constants of the transition layers created in this way are much closer to those of amorphous silicon (see e.g. [20]). Thus, one can expect that the transition layers are probably formed by damaged layers under the surfaces of the silicon single crystal substrates (for details see [20,40,45]). In Fig. 2 the spectral dependencies of the associated ellipsometric parameters $I_\textrm {s}$, $I_\textrm {c}$ and $I_\textrm {n}$ belonging to the silicon single crystal substrate covered by the transition layer are introduced. It is evident that the fits of the experimental data are very good which supports a correctness of the values of the optical parameters determined for the transition layer.

In the next step of the optical characterization of the SiO$_x$C$_y$H$_z$ thin film, the values of the thickness and optical constants of the transition layer were fixed in the values determined in the foregoing step, i.e. before depositing the characterized SiO$_x$C$_y$H$_z$ thin film. Thus, only the parameters characterizing the deposited SiO$_x$C$_y$H$_z$ thin film were determined in this second step. The found values of the parameters describing the thickness non-uniformity of this film corresponding to the polynomial of the second order are summarized in Table 1. The uncertainties introduced in this table correspond to statistical uncertainties determined by the least-squares method. These values can be compared with the values obtained using ISR, which are also shown in this table. It is necessary to point out that within the processing of the ISR data, the values of the spectral dependencies of the optical constants of the SiO$_x$C$_y$H$_z$ thin films were fixed in the values determined using spectroscopic ellipsometry. This means that the values of the local thicknesses corresponding to individual pixels were only parameters determined using ISR. On the basis of these local thicknesses the values of the coefficients of the quadratic polynomial were determined. The results in Table 1 imply that the differences in values of the corresponding parameters of the thickness non-uniformity determined using spectroscopic ellipsometry and ISR are relatively small.

Table 1. Values of the parameters describing the thickness non-uniformity of the characterized film of SiO$_x$C$_y$H$_z$.

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In Table 2, the values of the dispersion parameters of the Campi-Coriasso model are introduced for the upper and lower boundaries. The differences in values of the dispersion parameters corresponding to the upper boundary and lower boundary are relatively large which implies relatively large differences between the optical constants at the upper and lower boundaries, which is seen in Fig. 3.

Fig. 3. Spectral dependencies of the optical constants corresponding to the upper boundary and lower boundary (left) and the profiles of the optical constants of the SiO$_x$C$_y$H$_z$ thin film for photon energy $E=3.5$ eV (right).

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Table 2. Values of the dispersion parameters of the SiO$_x$C$_y$H$_z$ thin film (left). Values of quantity $\chi$ for the individual total corrections in the formulae for the reflection coefficients of the inhomogeneous SiO$_x$C$_y$H$_z$ thin film (right).

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In Fig. 3, the profiles of the optical constants of the SiO$_x$C$_y$H$_z$ thin film are plotted at photon energy $E=3.5$ eV. From Fig. 3 it can be seen that this film is absorbing for $E\ge 2.5$ eV in the region close to the upper boundary.

The quality of the fits of experimental data is expressed by means of the quantity $\chi$ defined as $\chi = (\Sigma /N_\textrm {exp})^{1/2}$, where symbols $\Sigma$ and $N_\textrm {exp}$ denote the weighted residual sum of squares and number of all measured values, respectively. If the weights in the least squares method were determined using the true statistical errors of measured values, then the ideal fit would give $\chi$ close to unity. Values of $\chi$ considerably larger than unity correspond to worse fits. However, we use only estimates for the measurement errors, thus, the optimal fit could correspond to value of $\chi$ different from unity, thus, the quantity $\chi$ can be used only to compare qualities of fits achieved using different models. In Table 2, the values of $\chi$ corresponding to the fits obtained by means of the formulae for the reflection coefficients $r_q$ containing the WKBJ approximation plus additional corrections are summarized (see Eq. (4)). Moreover, the value of $\chi$ achieved using the wedge-shaped thickness non-uniformity model corresponding to local thicknesses given by the first-order polynomial is introduced in this table. From Table 2, it is clear that the reflection coefficients for the characterized inhomogeneous thin film calculated using the formulae containing the term of the WKBJ approximation plus the term of the total correction of the first order are sufficient, i.e. it holds that $r_q = r_{0q} + \Delta r_{1q}$. This implies that the gradients of the profiles of the refractive index and extinction coefficient are rather small. If the gradients were larger, the terms corresponding to the total corrections of the higher orders would have to be used for expressing the reflection coefficient in Eq. (4) (see [46]). Furthermore, the values summarized in Table 2 imply that the wedge-shaped thickness non-uniformity is not suitable approximation for the non-uniformity of the studied film because the corresponding value of $\chi$ is relatively large.

The values of the coefficients in the quadratic polynomial determined by ellipsometry presented in Table 1 enable us to determine the profiles of local thicknesses corresponding to the cross-sections of the surface forming the upper boundary. In Fig. 4, the local thickness profiles of the film are plotted for the cross-sections with the plane of incidence and the rotations corresponding to angles $\beta =0^{\circ }$, $90^{\circ }$, $\beta =45^{\circ }$ and $\beta =135^{\circ }$. The local thickness profiles determined using ISR are also introduced in this figure for comparison. The ISR profiles of the local thicknesses were determined from the map of local thicknesses. One can see that there are certain differences in the corresponding profiles determined by ellipsometry and ISR. However, these differences are acceptable owing to the very complicated structure of the studied thin film and different systematic errors of both the optical techniques employed. Thus, the curves presented in Fig. 4 support a correctness of the results concerning thickness non-uniformity of the film achieved by ellipsometry. In Figs. 5 and 6, the maps of local thicknesses and the views on the three dimensional surface of the upper boundary of the characterized film of SiO$_x$C$_y$H$_z$ determined by ellipsometry and ISR are presented. These figures show that the surface representing the upper boundary of the selected film was determined correctly by the ellipsometric method described here, which is also supported by the values of the parameters presented in Table 1.

Fig. 4. Profiles of the local thicknesses of the thin film determined by ellipsometry and ISR for the four orientations of the sample.

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Fig. 5. Maps of local thicknesses of the thin film determined by ellipsometry (left) and ISR (right).

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Fig. 6. Three-dimensional views on the surface of the upper boundary of the thin film determined by ellipsometry (left) and ISR (right).

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The fits of the spectral dependencies of the associated ellipsometric parameters measured for incidence angle of $70^{\circ }$ and azimuth angle $\beta =0^{\circ }$ are plotted in Fig. 7. These fits are evidently very good. This supports the correctness of all the results obtained for the selected film of SiO$_x$C$_y$H$_z$ by the ellipsometric method. It should be noted that if the transition layer is not included in the stuctural model, then the quality of the fit of the experimental data is given by the value $\chi =7.6$ which corresponds to much worse result than the value $5.4$ obtained if the transition layer is taken into account.

Fig. 7. Spectral dependencies of the associated ellipsometric parameters at incidence angle $70^{\circ }$ (left) and the degree of polarization at the selected angles of incidence (right) for angle $\beta =0^{\circ }$: points represent experimental values, curves denote theoretical data.

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From the spectral dependencies of the degree of polarization, which are shown for three selected angles of incidence and the azimuth angle $\beta =0^{\circ }$ in Fig. 7, it is seen that considerable depolarization occurs in the light reflected from the characterized film. This depolarization is mainly caused by the thickness non-uniformity of this film. The depolarization does not occur for photon energies $E\ge 4.5$ eV since the film is not transparent in this region, therefore, the thickness non-uniformity has no influence on reflected light.

The results presented show that the method of spectroscopic ellipsometry described in this paper enables us to perform the complete optical characterization of the inhomogeneous SiO$_x$C$_y$H$_z$ thin film exhibiting the complicated thickness non-uniformity and the transition layer at the lower boundary of this film.

Similar results were obtained for the other samples of the inhomogeneous non-uniform polymer-like thin films, i.e. thin films of SiO$_x$C$_y$H$_z$, prepared by the technological procedure mentioned above.

6. Conclusion

In this paper, the method of spectroscopic ellipsometry enabling the complete optical characterization of inhomogeneous thin films exhibiting thickness non-uniformity and transition layers at their lower boundaries is presented. The method is illustrated by performing the complete characterization of the selected sample of the SiO$_x$C$_y$H$_z$ thin film prepared by PECVD onto silicon single crystal substrate. The method is divided into two steps. Within the first step the ellipsometric characterization of the transition layer originated during pretreating silicon substrate in Ar discharge is realized. In this way the thickness and spectral dependencies of the optical constants of the transition layer are determined. The values of the optical parameters of the transition layer are then fixed in the second step of the ellipsometric method. Within this second step the values of all the parameters characterizing the SiO$_x$C$_y$H$_z$ thin film are determined. The Campi–Coriasso model is utilized for determining the optical constants of the polymer-like thin film. The inhomogeneity is modeled by considering two sets of dispersion parameters for the SiO$_x$C$_y$H$_z$ thin film, one for the optical constants at the upper boundary and one for the optical constants at the lower boundary. The optical constants inside the SiO$_x$C$_y$H$_z$ thin film are then determined using linear interpolation of these dispersion parameters. The values of the parameters describing the thickness non-uniformity are determined simultaneously with the optical constants of the SiO$_x$C$_y$H$_z$ thin film. The optical constants of the transition layer are described using the combination of two Campi–Coriasso terms and exponential tail.

It was found that the formulae based on the multiple-beam interference model employing WKBJ approximation plus the total correction of the first order were sufficient for expressing the reflection coefficients of the selected inhomogeneous polymer-like film. This implies that the gradients of the refractive index and extinction coefficient profiles are relatively small even though the differences between these quantities at the upper and lower boundaries are rather large. This is caused by large values of the local thicknesses. It was also shown that the model of the thickness non-uniformity with local thicknesses given by the polynomial with at most quadratic terms in the coordinates along the sample surface describes the studied film very well, while the model of wedge-shaped thickness non-uniformity, which utilizes only the first-degree polynomial, is not sufficient.

The results obtained by ellipsometry for the thickness non-uniformity of the characterized film are supported by means of the map of the local thicknesses determined by ISR.

If the transition layer at the lower boundary is not taken into account in the structural model, inadequate fits of the experimental data are obtained. Thus, the transition layer must be considered in the optical characterization of the polymer-like thin films prepared by the technological procedure utilized.

The very good fits of the experimental ellipsometric data achieved in the second step of the method imply that the dispersion and structural models of the inhomogeneous non-uniform SiO$_x$C$_y$H$_z$ thin film were chosen correctly.

The method utilizing variable angle spectroscopic ellipsometry was used to perform the complete optical characterization of inhomogeneous thin film exhibiting general thickness non-uniformity and transition layer. Thus, the presented ellipsometric method extends our possibilities in the optical characterization of inhomogeneous thin films occurring in the fundamental research, applied research and innovations.

Funding

Ministerstvo Školství, Mládeže a Tělovýchovy (LM2018097); Grantová Agentura České Republiky (GACR 19-15240S); Vysoké Učenní Technické v Brně (FSI-S-20-6353).

Disclosures

The authors declare no conflicts of interest.

References

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		ellipsometry	ISR
$h_{0}$	[nm]	$960.7 \pm 1.1$	$959.5 \pm 0.3$
$h_{x}$	[nm]	$9.7 \pm 0.1$	$14.78 \pm 0.06$
$h_{y}$	[nm]	$9.5 \pm 0.1$	$11.10 \pm 0.06$
$h_{xx}$	[nm]	$2.01 \pm 0.09$	$2.73 \pm 0.03$
$h_{xy}$	[nm]	$1.35 \pm 0.09$	$1.30 \pm 0.03$
$h_{yy}$	[nm]	$1.69 \pm 0.09$	$2.23 \pm 0.03$

parameter		upper boundary	lower boundary		$χ$
$N$	[eV $^{2}$ ]	$1028 \pm 42$	$591 \pm 54$	WKBJ	5.813
$E_{g}$	[eV]	$2.41 \pm 0.01$	$8.05 \pm 0.09$	with term 1	5.419
$E_{c}$	[eV]	$15.4 \pm 0.2$	$9.1 \pm 0.1$	with terms 1+2	5.418
$B$	[eV]	$28 \pm 1$	$18 \pm 2$	with terms 1+2+3	5.418
				wedge	5.807

		ellipsometry	ISR
$h_{0}$	[nm]	$960.7 \pm 1.1$	$959.5 \pm 0.3$
$h_{x}$	[nm]	$9.7 \pm 0.1$	$14.78 \pm 0.06$
$h_{y}$	[nm]	$9.5 \pm 0.1$	$11.10 \pm 0.06$
$h_{xx}$	[nm]	$2.01 \pm 0.09$	$2.73 \pm 0.03$
$h_{xy}$	[nm]	$1.35 \pm 0.09$	$1.30 \pm 0.03$
$h_{yy}$	[nm]	$1.69 \pm 0.09$	$2.23 \pm 0.03$

parameter		upper boundary	lower boundary		$χ$
$N$	[eV $^{2}$ ]	$1028 \pm 42$	$591 \pm 54$	WKBJ	5.813
$E_{g}$	[eV]	$2.41 \pm 0.01$	$8.05 \pm 0.09$	with term 1	5.419
$E_{c}$	[eV]	$15.4 \pm 0.2$	$9.1 \pm 0.1$	with terms 1+2	5.418
$B$	[eV]	$28 \pm 1$	$18 \pm 2$	with terms 1+2+3	5.418
				wedge	5.807

Ellipsometric characterization of inhomogeneous thin films with complicated thickness non-uniformity: application to inhomogeneous polymer-like thin films

Abstract

1. Introduction

2. Sample preparation and experimental arrangement

2.1 Sample preparation

2.2 Experimental arrangement

3. Theoretical background

3.1 Structural model

3.2 Dispersion model

3.3 Reflection coefficients of inhomogeneous non-uniform thin films

3.4 Ellipsometric parameters of the inhomogeneous non-uniform thin films

3.5 Numerical calculations of the integrals

4. Data processing

5. Results and discussion

6. Conclusion

Funding

Disclosures

References

Cited By

Figures (7)

Tables (2)

Equations (18)

Optics Express